686 Appendix C • Answers & Hints for Selected Exercises
( ) e J" (n+1)"^1 sin x xi d x = L.; .;:.... Jk11: r(k+l)" I sin x xi d x -> .;:.... L.; Jk" r(k+l)" _1_dx 2(k+l)"
k=l k=l
n n
= 2 1 L.; "" k+1 k > "" L.; k+1^1 ---+ +oo.
k=l k=l- Va< c < d < b, l: f + l: f =(I: f + J: f) + (t f + l: f) = J: f + l: f.
- (a) Assume the hypotheses and define F(x) = l: f, G(x) = l: g on (-oo, b].
Then F, Gare monotone decreasing and G(x) ~ F(x) on (-oo, b]. x~lim -oo G(x)
exists, so by Exercise 4.4-B.15, G is bounded above on (-oo, b] and 1~ 00 g =
lim G(x) = sup{G(x): x::; b}. Thus, lim F(x)::; lim G(x) = 1~ 00 g.
x~-oo x~-oo x~-oo
(b) Suppose that Vx ::; b, 0 ~ f(x) ~ g(x). If 1~ 00 g converges, so does
1~ 00 f and 1~ 00 f ~ l~oo g.EXERCISE SET 7.9- (a) Obvious from Defn. 7.9.2.
(b) Suppose A, B have measure 0 and e: > 0. Then :3 co llections of open
00 00
intervals {In: n EN} and {Jn: n EN} such that As;,; LJ In and B s;,; LJ Jn
n=l n=l
and I: l(In) < e:/2 and I: l(Jn) < e:/2. Define the open intervals Kn= In/2 if n
00
is even and Kn= I(n+l)/2 if n is odd. Then AU B s;,; LJ Kn and I: l(Kn) < e:.
n=l
( c) Use (b) and mathematical induction.( d) Suppose { Ak : k E N} is a countable collection of sets of measure 0, and
let e: > 0. Then Vk E N, :3 collection {hn : n E N} of open intervals 3 Ak s;,;
00 00
LJ Ikn and I: l(Ikn) < e:/2k. Use the diagonal scheme shown in Thm 2.8.5
n=l n=l
to arrange this collection of intervals into a sequence {In} of open intervals.
Then kgl Ak s;,; kgl CQl hn) = nQl In. Since k~l l(h) ::; k~l c~l l(hn)) <
00 00
I: e:/2k = e:, LJ Ak has measure 0.
k=l k=l- Let A = {x E [a, b] : f(x) =J g(x)}, and Vn E N, let An = {x E [a, b]
00
lf(x) -g(x)I ~ 1/n}. Then x EA{::} ::Jn EN 3 x E An. Thus, A= LJ An· By
n=l
Exercises 7.9.l and 7.9.2, this set has measure 0. - Suppose f is integrable on [a, b], and define F(x) = l: f. By FTC-II,
F'(x) = f(x) at every point of [a, b] except possibly at a, band points where f is